Behaviors of entropy on finitely generated groups

Abstract : A variety of behaviors of entropy functions of random walks on finitely generated groups is presented, showing that for any $1-2 \leq \alpha \leq \beta \leq 1$, there is a group $\Gamma$ with measure $\mu$ equidistributed on a finite generating set such that $\liminf \frac{\log H {\Gamma,\mu}n}{\log n}=\alpha$ and $\limsup \frac{\log H {\G,\m}n}{\log n}=\beta$. The groups involved are finitely generated subgroups of the group of automorphisms of an extended rooted tree. The return probability and the drift of a simple random walk $Y n$ on such groups are also evaluated, providing an exemple of group with return probability satisfying $\liminf \frac{\log |\log PY n= \Gamma 1|}{\log n}=1-3$ and $\limsup \frac{\log |\log PY n= \G1|}{\log n}=1$ and drift satisfying $\liminf \frac{\log E||Y n||}{\log n}=1-2$ and $\limsup \frac{\log \E||Y n||}{\log n}=1$.